Difference between revisions of "Simulation in the aTAM"
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==Informal Definition of Simulation== | ==Informal Definition of Simulation== | ||
+ | ==Formal Definition== | ||
+ | |||
+ | From this point on, let $T$ be a $d$-dimensional tile set, and let %, for some $d \in \{2,3\}$. | ||
+ | $m\in\Z^+$. | ||
+ | An \emph{$m$-block supertile} over $T$ is a partial function $\alpha : \Z_m^d \dashrightarrow T$, where $\Z_m = \{0,1,\ldots,m-1\}$. Note that the dimension of the $m$-block is implicitly defined by the dimension of $T$. | ||
+ | Let $B^T_m$ be the set of all $m$-block supertiles over $T$. | ||
+ | The $m$-block with no domain is said to be $\emph{empty}$. | ||
+ | For a general assembly $\alpha:\Z^d \dashrightarrow T$ and $(x_0,\ldots x_{d-1})\in\Z^d$, define $\alpha^m_{x_0,\ldots x_{d-1}}$ to be the $m$-block supertile defined by $\alpha^m_{x_0,\ldots, x_{d-1}}(i_0,\ldots, i_{d-1}) = \alpha(mx_0+i_0,\ldots, mx_{d-1}+i_{d-1})$ for $0 \leq i_0, \ldots, i_{d-1}< m$. | ||
+ | %\paragraph{Block replacement function.} | ||
+ | For some tile set $S$ of dimension $d' \geq d$, a partial function $R: B^{S}_m \dashrightarrow T$ is said to be a \emph{valid $m$-block supertile representation} from $S$ to $T$ if for any $\alpha,\beta \in B^{S}_m$ such that $\alpha \sqsubseteq \beta$ and $\alpha \in \dom R$, then $R(\alpha) = R(\beta)$. | ||
==References== | ==References== |
Revision as of 11:25, 12 July 2013
Informal Definition of Simulation
Formal Definition
From this point on, let \(T\) be a \(d\)-dimensional tile set, and let %, for some \(d \in \{2,3\}\). \(m\in\Z^+\). An \emph{\(m\)-block supertile} over \(T\) is a partial function \(\alpha : \Z_m^d \dashrightarrow T\), where \(\Z_m = \{0,1,\ldots,m-1\}\). Note that the dimension of the \(m\)-block is implicitly defined by the dimension of \(T\). Let \(B^T_m\) be the set of all \(m\)-block supertiles over \(T\). The \(m\)-block with no domain is said to be \(\emph{empty}\). For a general assembly \(\alpha:\Z^d \dashrightarrow T\) and \((x_0,\ldots x_{d-1})\in\Z^d\), define \(\alpha^m_{x_0,\ldots x_{d-1}}\) to be the \(m\)-block supertile defined by \(\alpha^m_{x_0,\ldots, x_{d-1}}(i_0,\ldots, i_{d-1}) = \alpha(mx_0+i_0,\ldots, mx_{d-1}+i_{d-1})\) for \(0 \leq i_0, \ldots, i_{d-1}< m\). %\paragraph{Block replacement function.} For some tile set \(S\) of dimension \(d' \geq d\), a partial function \(R: B^{S}_m \dashrightarrow T\) is said to be a \emph{valid \(m\)-block supertile representation} from \(S\) to \(T\) if for any \(\alpha,\beta \in B^{S}_m\) such that \(\alpha \sqsubseteq \beta\) and \(\alpha \in \dom R\), then \(R(\alpha) = R(\beta)\).